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Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).
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%I #22 Jun 21 2021 03:01:13

%S 1,1,1,1,0,2,1,0,1,3,1,0,0,1,5,1,0,0,1,1,8,1,0,0,0,1,2,13,1,0,0,0,1,0,

%T 2,21,1,0,0,0,0,1,1,3,34,1,0,0,0,0,1,0,2,4,55,1,0,0,0,0,0,1,0,1,5,89,

%U 1,0,0,0,0,0,1,0,1,1,7,144,1,0,0,0,0,0,0,1,0,2,3,9,233

%N Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

%C A(n,k) is the number of compositions of n into parts k and k+1.

%H Seiichi Manyama, <a href="/A306713/b306713.txt">Antidiagonals n = 0..139, flattened</a>

%F A(n,k) = Sum_{j=0..floor(n/k)} binomial(j,n-k*j).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 2, 1, 0, 0, 0, 0, 0, 0, 0, ...

%e 3, 1, 1, 0, 0, 0, 0, 0, 0, ...

%e 5, 1, 1, 1, 0, 0, 0, 0, 0, ...

%e 8, 2, 0, 1, 1, 0, 0, 0, 0, ...

%e 13, 2, 1, 0, 1, 1, 0, 0, 0, ...

%e 21, 3, 2, 0, 0, 1, 1, 0, 0, ...

%e 34, 4, 1, 1, 0, 0, 1, 1, 0, ...

%e 55, 5, 1, 2, 0, 0, 0, 1, 1, ...

%t T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jun 21 2021 *)

%Y Columns 1-10 give A000045(n+1), A182097, A017817, A017827, A017837, A017847, A017857, A017867, A017877, A017887.

%Y Cf. A306646, A306680.

%K nonn,tabl

%O 0,6

%A _Seiichi Manyama_, Mar 05 2019